Question 1

The test statistic that is used when testing a null hypothesis for a population variance is the standard normal z-value.

Accepted Answer

The test statistic used when testing a null hypothesis for a population variance is the chi-square ($\chi^2$) statistic, not the standard normal z-value.

Question 2

In a two-tailed hypothesis test for the difference between two population variances, if s1 = 3 and s2 = 5, then the test statistic is F = 2.7778.

Accepted Answer

The test statistic for the difference between two population variances is F = s1^2 / s2^2, so F = 3^2 / 5^2 = 2.7778.

Question 3

Assume a sample of size n = 12 has been collected. To perform a hypothesis test of a population variance using a 0.05 level of significance, where the null hypothesis is: H0 : σ2 = 25 The upper tail critical value is 21.92.

Accepted Answer

The upper tail critical value is 21.92, which is greater than the sample variance of 25. Therefore, the null hypothesis is rejected and the alternative hypothesis is accepted.

Question 4

The managers for a fruit juice facility claim that the standard deviation for the ounces per bottle on the new automated line is not the same as the older manual line. Given this, the correct null and alternative hypotheses for performing the statistical test are: H0 : σ1 = σ2 Ha : σ1 ≠ σ2

Accepted Answer

The null hypothesis (H0) states there is no difference in standard deviations between the two lines (σ1 = σ2), and the alternative hypothesis (Ha) states there is a difference (σ1 ≠ σ2), which correctly reflects the managers' claim.

Question 5

In a hypothesis test involving two population variances, if the null hypothesis states that the two variances are strictly equal, then the test statistic is a chi-square statistic.

Accepted Answer

The test statistic for comparing two population variances is typically the F-statistic, derived from the F-distribution, not the chi-square statistic.

Question 6

In a two-tailed hypothesis test for the difference between two population variances, if s1 = 3 and s2 = 5, then the test statistic is F = 1.6667.

Accepted Answer

The test statistic F is calculated as the ratio of the two sample variances (s1^2/s2^2 or s2^2/s1^2), not the standard deviations.

Question 7

A contract calls for the strength of a steel rod to stand up to pressure of 200 lbs per square inch on average. The contract also requires that the variability in strength for individual steel rods be no more than 5 pounds per square inch. If a random sample of n = 15 rods is selected and the sample standard deviation is 6.7 pounds, the test statistic is approximately χ2 = 25.138.

Accepted Answer

The test statistic for checking the variability (standard deviation or variance) of a population, given a sample, is indeed calculated using the chi-square (χ2) distribution formula: χ2 = (n-1)s^2 / σ^2, where n is the sample size, s is the sample standard deviation, and σ^2 is the population variance. Given the sample standard deviation of 6.7 pounds per square inch and the requirement that the variability not exceed 5 pounds per square inch, the calculation of the test statistic would involve these values, aligning with the provided χ2 value, assuming the correct formula and population variance (σ^2) were used in the calculation.

Question 8

The central location and shape of the chi-square distribution depend only on the population variance.

Accepted Answer

The central location and shape of the chi-square distribution depend on the degrees of freedom, not the population variance.

Question 9

One of the most important aspects of quality improvement is the idea of reducing the variability in a product or service. For instance, a major bank has worked to reduce the variability in the service time at the drive-through. The managers believe that the standard deviation in service time should not exceed 30 seconds. To test whether this goal is being achieved, a random sample of n = 25 cars is selected each week and the service time for each car is measured. Last week, the mean time was 345 seconds with a standard deviation equal to 38 seconds. Given this information, if the significance level is 0.10, the critical value from the chi-square table is about 34.3.

Accepted Answer

The critical value from the chi-square table for a given significance level and degrees of freedom (which in this case would be n-1 = 24 for a sample size of 25) is not specified in the question. The value of 34.3 mentioned does not directly relate to the significance level without specifying the degrees of freedom and whether it's a one-tailed or two-tailed test. Chi-square critical values depend on the degrees of freedom and the significance level, and without this specific information, the exact critical value cannot be determined.

Question 10

A one-tailed hypothesis test for a population variance always has the rejection region in the upper tail.

Accepted Answer

A one-tailed hypothesis test for a population variance always has the rejection region in the upper tail because the alternative hypothesis is always that the population variance is greater than the hypothesized value.

Question 11

A machine that is used to fill soda pop cans with pop has an adjustable mean fill setting, but the standard deviation is not supposed to exceed 0.18 ounces. To make sure that this is the case, the managers at the beverage company each day select a random sample of n = 6 cans and measure the fill volume carefully. In one such case, the following data (ounces per can) were observed.  Based on these sample data, the test statistic is approximately χ2 = 5.01.

Accepted Answer

The answer of A machine that is used to fill...

Question 12

In a hypothesis test for the equality of two variances, the lower-tail critical value does not need to be found as long as the larger sample variance is placed in the denominator of the test statistic.

Accepted Answer

The answer of In a hypothesis test for the equality...

Question 13

When using a chi-square test for the variance of one population, we are assuming that the population is normally distributed.

Accepted Answer

The chi-square test for the variance of one population assumes that the population from which the sample is drawn is normally distributed, as the test's validity depends on this assumption.

Question 14

A potato chip manufacturer has two packaging lines and wants to determine if the variances differ between the two lines. They take a sample of n= 15 bags from each line and find the following: The value of the test statistic is F = 1.5

Accepted Answer

The statement does not provide enough information to determine if the variances differ. To conclude, one would need to compare the test statistic to a critical value from the F-distribution table, considering the degrees of freedom for both samples.

Question 15

In a two-tailed hypothesis test for the difference between two population variances, the test statistic is an F-ratio formed by putting the larger sample variance in numerator.

Accepted Answer

In a two-tailed hypothesis test for the difference between two population variances, the F-ratio is calculated by dividing the larger sample variance by the smaller sample variance, placing the larger variance in the numerator to ensure the F-ratio is greater than or equal to 1.

Question 16

The F-distribution is used to test whether two sample variances are equal.

Accepted Answer

The F-distribution is indeed used in statistics to compare two sample variances and test if they are equal, typically in the context of an ANOVA test.

Question 17

If we are interested in performing a one-tailed, upper-tail hypothesis test about a population variance where the level of significance is .01 and the sample size is n = 25, the critical chi-square value to be used is 42.9798.

Accepted Answer

For a one-tailed, upper-tail hypothesis test about a population variance with a significance level of .01 and a sample size of n = 25, degrees of freedom would be n - 1 = 24. Looking up the critical value in a chi-square distribution table for 24 degrees of freedom at the .01 significance level indeed gives a critical value of approximately 42.9798.

Question 18

For a given significance level, increasing the sample size will make the chi-square distribution more skewed.

Accepted Answer

Increasing the sample size tends to make the chi-square distribution less skewed, approaching a normal distribution as the sample size increases.

Question 19

The variance in the diameter of a bolt should not exceed 0.500 mm. A random sample of n = 12 bolts showed a sample variance of 0.505 mm. The test statistic is χ2 = 11.11.

Accepted Answer

The test statistic is χ2 = 11.11, which is greater than the critical value of χ2 = 9.488 (for α = 0.05 and df = 11). Therefore, we reject the null hypothesis and conclude that the variance in the diameter of a bolt exceeds 0.500 mm.

Question 20

In a two-tailed test for the equality of two variances, the critical value is determined by going to the F- distribution table with an upper-tail area equal to alpha divided by two.

Accepted Answer

In a two-tailed test for the equality of two variances, the critical value is indeed determined by consulting the F-distribution table with an upper-tail area equal to the significance level (alpha) divided by two, because the test is checking for deviations in both directions from the null hypothesis.

Quiz 11: Hypothesis Tests and Estimation for Population Variances